The Astrophysical Journal, 684:212Y225, 2008 September 1
# 2008. The American Astronomical Society. All rights reserved. Printed in U.S.A.
GRAVITATIONAL INSTABILITY IN RADIATION PRESSUREYDOMINATED BACKGROUNDS
Todd A. Thompson
Department of Astronomy and Center for Cosmology and Astroparticle Physics, The Ohio State University, Columbus, OH 43210;
[email protected]
Received 2007 November 21; accepted 2008 April 11
ABSTRACT
I consider the physics of gravitational instabilities in the presence of dynamically important radiation pressure and
gray radiative diffusion. For any nonzero radiation diffusion rate on an optically thick scale, the medium is unstable
unless the classical gas-only isothermal Jeans criterion is satisfied. When diffusion is ‘‘slow,’’ even though the Jeans
instability is stabilized by radiation pressure on scales smaller than the adiabatic Jeans length, on these same spatial scales
the medium is unstable to a diffusive mode. In this regime, neglecting gas pressure, the characteristic growth timescale is
independent of spatial scale and given by (3cs2 )/(4Gc), where cs is the adiabatic sound speed. This timescale is that
required for a fluid parcel to radiate away its thermal energy content at the Eddington limit, the Kelvin-Helmholz
timescale for a radiation pressureYsupported self-gravitating object. In the limit of ‘‘rapid’’ diffusion, radiation does
nothing to suppress the Jeans instability and the medium is dynamically unstable unless the gas-only Jeans criterion
is satisfied. I connect with treatments of Silk damping in the early universe. I discuss several applications, including
photons diffusing in regions of extreme star formation (starburst galaxies and parsec-scale AGN disks), and the diffusion
of cosmic rays in normal galaxies and galaxy clusters. The former (particularly, starbursts) are ‘‘rapidly’’ diffusing and
thus cannot be supported against dynamical instability in the linear regime by radiation pressure alone. The latter are
more nearly ‘‘slowly’’ diffusing. I speculate that the turbulence in starbursts may be driven by the dynamical coupling
between the radiation field and the self-gravitating gas, perhaps mediated by magnetic fields, and that this diffusive
instability operates in individual massive stars.
Subject headinggs: galaxies: starburst
1. INTRODUCTION
1954, 1961; Mestel 1965; Lynden-Bell 1966) and the physics of
radiating flows (e.g., Mihalas & Mihalas 1983, 1984, and references therein; Spiegel 1957; Field 1971; Kaneko et al. 1976, 2000;
Bisnovatyi- Kogan & Blinnikov 1978, 1979; Dzhalilov et al. 1992;
Zhugzhda et al. 1993; Arons 1992; Bogdan et al. 1996; Gammie
1998; Blaes & Socrates 2001, 2003; Socrates et al. 2005), there has
been relatively little work on self-gravitating environments where
radiation might play an important dynamical role (however, see
Kaneko & Morita 2006; Vranjes & Cadez 1990; Vranjes 1990).
Perhaps the first and most familiar treatment of self-gravitating
radiation pressureYdominated media was carried out by Silk (1967,
1968), and then extended by Peebles & Yu (1970) and Weinberg
(1971), in the context of acoustic wave damping of primeval
fluctuations by radiative diffusion—‘‘Silk damping’’ (see also
Hu & Sugiyama 1996; Dodelson 2003). However, these works
focus specifically on the damping rate of acoustic fluctuations and
the generation of entropy and did not delineate how the Jeans
criterion is modified on scales larger than the gas-only Jeans length
when radiation is dynamically dominant and diffusing. They also
do not discuss the physics of slow nondynamical diffusive modes.
This paper is motivated by astrophysical systems where selfgravity and radiation are essential. These include sites of extreme
massive star formation such as compact starburst galaxies and
the parsec-scale disks or obscuring ‘‘tori’’ thought to attend the
process of fueling active galactic nuclei. These environments are
marked by high radiation energy density and high gas density, as
well as optical depths to their own dust-reprocessed infrared radiation that may significantly exceed unity (x 3 and, e.g., Pier &
Krolik 1992; Goodman 2003; Sirko & Goodman 2003; Thompson
et al. 2005, hereafter TQM05; Chang et al. 2007). In each of these
systems radiation pressure can be comparable to gravity, and the
associated photon energy density is rivaled only by the energy
density in turbulence and, potentially, the contributions from
Standard treatments of the Jeans instability assume the medium
is homogeneous and isotropic and governed by a barotropic equation of state. Employing the ‘‘Jeans swindle’’ so that the Poisson
equation is satisfied in an ad hoc way with no background gradients in density, the dispersion relation
! 2 ¼ cg2 k 2 4G
ð1Þ
follows from a linear analysis. Here, cg is the gas sound speed
and is the mass density. The Jeans instability is long wavelength;
for scales larger than the Jeans length,
2kJ1 ¼ kJ ¼ cg (/G)1=2 ;
ð2Þ
the system is dynamically unstable under the action of a perturbation to the density and the attending increase in the gravitational
potential (Jeans 1928; e.g., Binney & Tremaine 1987). On scales
smaller than kJ the medium responds to a compression with a
restoring pressure force. Equation (2) can be obtained by equating
the acoustic sound crossing timescale on a scale kJ with the dynamical timescale. Equivalently, equation (2) may be read as expressing the fact that for stability, the total thermal energy of the
medium within a volume k3J must exceed the gravitational potential energy. When the system is unstable, it continuously and spontaneously transitions to states of lower total energy by liberating
thermal energy (e.g., Chandrasekhar 1961).
The purpose of this paper is to understand how the classical gas
Jeans criterion is modified by radiation and to ask in which astrophysical environments such a modification might be important.
Although there are many treatments in the literature of both the
Jeans instability (e.g., Jeans 1928; Ledoux 1951; Chandrasekhar
212
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GRAVITATIONAL INSTABILITY
cosmic rays and magnetic fields.1 The very high radiation energy
densities in these systems led TQM05 to propose a theory of
marginally Toomre-stable radiation pressure supported starburst
and AGN disks (see also Scoville et al. 2001; Scoville 2003). In
addition, this analysis may be of some interest for the stability
of individual massive stars and for self-gravitating media whose
pressure is dominated by cosmic rays, either in normal star-forming
galaxies or galaxy clusters.
In x 2, I present a simple linear analysis of the gravitational
instability. Appendix A discusses uniform rotation in the background medium. More detailed treatments of radiation transport
are considered in Appendix B (see also Kaneko & Morita 2006).
In x 3, I discuss the relevance of the results derived for a number
of astrophysical environments and x 4 provides a summary.
2. GRAVITATIONAL INSTABILITY WITH RADIATION
Here, I describe the simplest nontrivial treatment of the Jeans
problem with radiation pressure and diffusion that captures the
physics needed for a qualitative understanding (see Appendix B
for a more detailed treatment). The equations express continuity,
momentum and total energy conservation, self-gravity, and equilibrium optically thick radiative diffusion (e.g., Mihalas & Mihalas
1984). They are
@
þ : = (v) ¼ 0;
@t
@v
1
þ v = :v ¼ :P :;
@t
@U
þ v = :U þ (U þ P): = v ¼ : = F;
@t
92 ¼ 4G;
c
:ur :
F¼
3
ð3Þ
the medium and radiation field as homogeneous, isotropic, and in
radiative equilibrium: ¼ const:, P ¼ const:, U ¼ const:, and
v ¼ : = F ¼ F ¼ 0. The perturbed equations are
ð6Þ
ð7Þ
Here U ¼ ug þ ur and P ¼ pg þ pr are the total internal energy
density and pressure, and the subscripts r and g refer to the radiation and the gas, respectively. The radiation pressure force F/c
is contained in the :P term in equation (4). In addition, pg ¼
k B T /mp , ur ¼ aT 4 ¼ 3pr , ug ¼ pg /( 1), is the gas mass
density, is the adiabatic index of the gas, F is the radiative flux,
and is the opacity.
For simplicity, I take constant and I do not distinguish between the Planck-, flux-, and Rosseland-mean opacities. The
above equations also neglect the time dependence of the radiation
field, and they assume that the radiation and gas temperatures are
exactly equal (see Appendix B; see also Mihalas & Mihalas 1984;
Gammie 1998; Blaes & Socrates 2003; Kaneko & Morita 2006).
Because the Eddington approximation has been made, the effects
of photon viscosity have been neglected (e.g., Weinberg 1971;
Agol & Krolik 1998). In addition, in considering extreme star
formation environments where radiation is reprocessed by dust,
the above equations neglect the two-fluid nature of the coupled
dust-gas system; that is, they assume perfect collisional and energetic coupling between the dust and gas (see x 3). Finally, no
terms representing sources of optically thin radiative heating or
cooling are included.
I consider perturbations of the form q ! q þ q exp (ik = x i!t), keep only linear terms, employ the Jeans swindle, and take
1
For a recent assessment of the strength of magnetic fields in starburst galaxies, see Thompson et al. (2006). For a discussion of cosmic-ray feedback in
galaxies, see Socrates et al. (2008).
ð8Þ
i!v þ ik(P=) þ ik ¼ 0;
ð9Þ
i!U þ (U þ P)ik = v þ ik = F ¼ 0;
ð10Þ
2
k 4G ¼ 0;
ð11Þ
ikur þ (3=c)F ¼ 0:
ð12Þ
The thermodynamic perturbations to the total pressure and energy
density are
@P @P þ
T ;
ð13Þ
P ¼
@ T
@T @U @U þ
T :
ð14Þ
U ¼
@ @T T
Note that (@pr /@)jT ¼ (@ur /@)jT ¼ 0 so that (@P/@)jT ¼
(@pg /@)jT ¼ cT2 —that is, only the gas makes a contribution to
=
the total isothermal sound speed, (@P/@jT )1 2 ¼ cT , in equation (13). The perturbation to the radiation energy density is
written as
ð4Þ
ð5Þ
i! þ ik = v ¼ 0;
ur ¼ 4aT 3 T ¼ AT ;
ð15Þ
where the last equality defines A ¼ @ur /@T . Combining these
thermodynamic relations with the perturbation equations, one
finds that
i!˜
þ cT2 cs2 k 2 ; ð16Þ
0 ¼ ! 2 cT2 k 2 þ 4G 1 þ
!
where
ck 2 A
!˜ ¼
3 CV
ð17Þ
is the diffusion rate on a scale k 1 , CV ¼ (@U /@T )j is the total
specific heat, cs2 ¼ (@P/@)js is the square of the adiabatic sound
speed for the gas and radiation, S is the total entropy, and the
identity
@P @P @P U þP
@U ¼
þ
ð18Þ
@ S @ T @U @ T
has been employed. Expanding the dispersion relation and combining terms, equation (16) becomes
! 3 þ i! 2 !˜ ! cs2 k 2 4G i!˜ cT2 k 2 4G ¼ 0: ð19Þ
Note that the dimensionless ratio
A
@ur
¼
CV
@T
!1 @U ug 1
¼ 1þ
@T 4ur
ð20Þ
that appears in equation (17) for !˜ approaches unity in the limit
ug /(4ur ) ! 0, and zero in the limit ug /(4ur ) ! 1. Therefore, as
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THOMPSON
2
ur /ug ! 0, cs2 ! cs;g
¼ cT2 , and (A/CV ) ! 0, equation (19)
reduces to
! 3 ! cT2 k 2 4G ¼ 0
ð21Þ
in the gas pressureYdominated limit, fully analogous to the classical Jeans criterion in equation (1), but includes the entropy
mode ! ¼ 0 (e.g., Lithwick & Goldreich 2001) and explicitly
2
¼ cT2 . In the opposite,
contains the adiabatic gas sound speed cs;g
2
¼
radiation pressureYdominated limit, ur /ug ! 1, cs2 ! cs;r
2
(4pr /3) 3 cT . Neglecting gravity, the dispersion relation for
radiation pressure acoustic waves under the assumption of optically
2 2
˜ cs;r
k ) 0,
thick equilibrium radiative diffusion, !(! 2 þ i!!
is obtained from equation (19).
2.1. Dimensionless Numbers
Three dimensionless numbers determine the character of the
modes admitted by equation (19). The first measures the importance of gas pressure alone in supporting the medium on a
scale k 1 :
#T ¼
cT2 k 2 =(4G):
ð22Þ
The criterion #T > 1 is the classical gas-only Jeans criterion for
gravitational stability (cf. eq. [2]); #T is the ‘‘isothermal Jeans
number.’’ The isothermal Jeans length follows by taking #T ¼ 1:
kJ;T ¼ 2=kJ;T ¼ 2cT =(4G)1=2 :
ð23Þ
The second and third dimensionless ratios combine to determine
the importance of radiation pressure. The first is
#s ¼ cs2 k 2 =(4G);
ð24Þ
the ‘‘adiabatic Jeans number,’’ analogous to #T , but which includes the contribution from radiation pressure. Taking #s ¼ 1
defines the adiabatic Jeans length:
kJ;s ¼ 2=kJ;s ¼ 2cs =(4G)1=2 :
ð25Þ
The second ratio is
¼
ck 2 A
1
;
3 CV (4G)1=2
ð26Þ
a measure of the diffusion rate. The limits of rapid ( 3 1)
and slow (T1) diffusion are considered in xx 2.2 and 2.3,
respectively.
In analogy with the classical Jeans criterion, one might guess
that if #s is larger than unity, then in the limit of slow diffusion
the medium is stable. This turns out to be false, as I show in x 2.3.
In fact, if #T < 1 on a scale k 1 , then the medium is unstable
regardless of #s .
Using 2 ¼ ! 2 /(4G), and the definitions for #T , #s , and ,
equation (19) can be written as
3
2
þ i (#s 1) i(#T 1) ¼ 0:
ð27Þ
2.2. Rapid Diffusion
In the limit of rapid diffusion ( 3 #s , #T , and 1), the three
roots of equation (27) are
(#s #T )
;
2
(#s #T )
i þ i
;
(#T 1)1=2 i
Vol. 684
to first order in 1 . When #T > 1, the roots in equation (28) correspond to stable radiation- and gravity-modified gas acoustic
waves. For large #T , these modes propagate at the isothermal
sound speed of the gas; large ensures isothermality. In the
limit #T ! 0 and 3 #s , equation (28) is simply i and
the medium is dynamically unstable. This is the classical gasonly isothermal Jeans instability. Note that the limit of rapid
diffusion in equation (28) is distinct from the high-k limit, because
at high-k gravity, which dictates stability/instability, disappears.
However, to make an apposite comparison with the literature I
take the high-k limit and for the acoustic modes I find that
2
3pg 2
ur 1 þ
;
! cT k i
3c
4ur
ð30Þ
in agreement with Blaes & Socrates (2003, their eq. [62]). Equation (29) corresponds to the purely damped radiation diffusion
wave.
2.3. Slow Diffusion
In the limit of slow diffusion ( T#s , #T , and 1),
i #s #T
1=2
(#s 1) ;
2 #s 1
#T 1
i
:
#s 1
ð31Þ
ð32Þ
If #s > 1, equation (31) corresponds to two stable damped
gravity-modified radiation acoustic waves.2 For #s 3 1 and
#s 3 #T , the damping rate for these radiation acoustic waves is
simply /2. Conversely, when #s < 1 (and, thus, #T < 1)—that
is, on scales larger than the adiabatic Jeans length (eq. [25])—
the medium is dynamically unstable to the Jeans instability:
! i.
Equation (32) is key. It says that there is an intermediate
range in spatial scale k 1 , larger than isothermal Jeans length
(eq. [23]) and smaller than the adiabatic Jeans length (eq. [25]),
that is always unstable. For negligible gas pressure (#T ! 0), it is
precisely when the adiabatic Jeans number #s is greater than unity
and the dynamical Jeans instability is suppressed in equation (31)
that the diffusive mode in equation (32) is unstable. Even for
arbitrarily large #s and small , if the classical gas-only Jeans
criterion indicates instability—that is, if #T < 1—then the medium is unstable.
In a highly radiation pressure-dominated medium with #T T
1T#s , this diffusive mode grows at a rate
!i
4G c
2
3 cs;r
ð33Þ
in the high-k limit, independent of spatial scale. This expression is easy to understand as the rate at which a self-gravitating
fluid parcel of volume V and mass M radiates its total thermal
energy content [e (4/3)ur V ] at the Eddington limit (ė ¼
4GMc/); it is the inverse of the Kelvin-Helmholz timescale:
1
2
ė/e 4Gc/(3cs;r
).
tKH
Equivalently, the only terms from the Euler and energy equations that contribute to this branch of the dispersion relation are
ð28Þ
ð29Þ
2
Note that the real part of is modified by at the level (#s =
1)1 2 (2 /2)(#s #T )(#s 1)3=2 in eq. (31) if the second-order term in is
kept.
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GRAVITATIONAL INSTABILITY
the approximate equalities kP/ k and (U þ P)k = v k = F (cf. eqs. [9] and [10]). Combining the energy equation
with the continuity equation, (/) ¼ ( k = v)/!, and assuming
that the medium is radiation pressure dominated,
ck 2
!¼i
pr
1 :
4 pr
ð34Þ
On the other hand, the approximate equality kP/ k
implies that
pr 4G
¼
:
k2
ð35Þ
Combining equations (34) and (35), one finds precisely equation (33). Substituting into the continuity equation, I find that
2 ck 2 pr
4G
c
k = v ¼ i
¼i
;
c
4pr
4pr
ð36Þ
which relates the density and velocity perturbations.
Therefore, in a radiation pressureYdominated medium with
slow diffusion (see x 2.5) the characteristic time for collapse on
scales smaller than the adiabatic Jeans length is tKH , not the dynamical timescale. Although the growth timescale becomes long
as and cs;r become large, for 6¼ 0 the medium is never formally stable if #T < 1. In addition, the existence of this instability does not require a pure radiation-only gas with adiabatic
index of 4/3. Written another way, the growth timescale for this
diffusive instability at large k is
2
tKH tdiA tdyn =tr ;
ð37Þ
in an optically thick, slowly diffusing, radiation pressureY
dominated medium, where tr ¼ (cs;r k)1 is the radiation
pressure acoustic sound crossing timescale, and tdiA ¼ 3/ck 2
=
and tdyn ¼ (4G)1 2 are the diffusion and dynamical timescales, respectively.
2.4. Criterion for Existence of the Radiation Acoustic Mode
If the adiabatic Jeans number is larger than unity (#s > 1),
then when the radiation acoustic sound crossing timescale on a
scale k 1 is shorter than the diffusion timescale across that same
spatial scale, the radiation acoustic mode can be supported by
the medium. Thus, there is a critical diffusion rate defined by the
1=2
, for which the radiation acoustic
rough inequality 1
c;r k #s
mode exists. This criterion on the diffusion rate can be obtained
from an approximate solution to equation (27) in the limit #s 31
and #T ! 0, obtained by setting 0 for the radiation acoustic
mode. I find that
c;r
2 c
1
;
ð38Þ
1 1=2 6 cs;r k
2#s
where k ¼ (2k 1 ) is the optical depth on a spatial scale
2k 1 . Thus, if is small with respect to c;r , then the diffusion
rate on a scale k 1 is small compared to cs;r k and the radiation
acoustic mode can be supported. Conversely, for k c;r , such
a mode does not exist. Note that the critical value c;r is decreased by nonzero #T (see eq. [31]) and modified if #s is larger
than, but near, unity.
2.5. Criterion Defining ‘‘Rapid’’ and ‘‘Slow’’ Diffusion
Sections 2.2 and 2.3 distinguish between the limits of ‘‘rapid’’
and ‘‘slow’’ diffusion. The criterion that separates these two limits
defines a critical diffusion rate c;diA that can be estimated by
setting the growth timescale for the unstable diffusion mode in
equation (32) equal to unity, the inverse of the dynamical timescale. When #s 31 3 #T ,
c;diA #s :
ð39Þ
For > c;diA the medium is ‘‘rapidly’’ diffusing, and for <
c;diA it is ‘‘slowly’’ diffusing. Alternatively, equation (39) may
be written as (cf. eq. [37])
1
ck 2 (4G)1=2
j tr (tdiA tdyn )1=2 ;
2 k2
3 cs;r
ð40Þ
if the radiation pressure acoustic sound crossing timescale on a
scale k 1 is less than the geometric mean between the diffusion
timescale on that same spatial scale and the dynamical timescale, then diffusion is ‘‘slow’’ and the stability properties of
the medium are best described by x 2.3. Conversely, if tr >
=
(tdiA tdyn )1 2 , diffusion is ‘‘rapid’’ (x 2.2). This criterion is valid
only at high-k and in that regime is independent of spatial scale.
Equations (38) and (39) imply that c;diA can be greater than c;r
and therefore that even though diffusion is ‘‘slow,’’ the radiation
acoustic mode is not supported.
2.6. Solutions to the Dispersion Relation
The limits of fast and slow diffusion, the criterion separating
them, and the range of existence of the acoustic modes and their
damping rates are illustrated in the solution to equation (27)
presented in Figures 1 and 2, which show the modes obtained for
a wide range of , at fixed #s and #T . Increasing while keeping
#s and #T constant can be thought of as a continuous decrease in
the opacity at fixed k 1 . Open and filled circles show the real
and imaginary part of , respectively. Individual pieces of the various roots are labeled for comparison with equations (28)Y(32).
The left panel of Figure 1 shows a case with #s > 1 and #T < 1.
The unstable mode is the only positive imaginary root. For large =
it is the dynamical Jeans instability: i(1 #T )1 2 , whereas
for small it is the diffusive mode of equation (32). The dotted
line shows the approximation to c;r (eq. [38]). Because for the
parameters chosen, c;r 2#1s =2 #s , the dotted line denoting
c;r also roughly corresponds to c;diA (eq. [39]). For P c;r ,
the radiation acoustic modes are evident and modestly damped.
For any 6¼ 0, the medium is unstable, because #T < 1. Note
that the purely damped mode i is off-scale for large (eq. [29]). Contrast the left panel of Figure 1 with the right panel,
which shows the same calculation, but with #T ¼ 3/2 > 1. Because #T > 1, the medium is stable for any . For the parameters
chosen, the adiabatic acoustic mode, which contains contributions
from radiation and gas, joins smoothly into the isothermal gas
acoustic mode at modest .
Figure 2 presents a similar calculation, but more radiation
pressure dominated, with #s ¼ 20 and #T ¼ 1/10 (left) and #T ¼
3/2 (right). The left panel of Figure 1 is qualitatively identical
to the left panel of Figure 2, but in the latter there is a clear
separation between c;r (dotted line) and c;diA (dashed line).
The right panel of Figure 2 again shows a case with #T > 1, so
that the Jeans instability is stabilized for any , but shows the
separation in between the adiabatic acoustic mode, which is
here highly radiation pressure dominated ( P c;r ), and the
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THOMPSON
Vol. 684
Fig. 1.—Left: Solution to eq. (27) for #s ¼ 3 and #T ¼ 1/2, for 103 103 at fixed k 1 . Open and filled circles show the real and imaginary parts of the three
roots , respectively. Positive complex roots indicate instability. Because #T < 1 on the scale chosen, the medium is unstable any 6¼ 0. For large the instability is
dynamical, whereas for small the medium is unstable to the diffusive instability given by eq. (32). The dotted line shows the approximate solution for c;r (eq. [38]),
=
below which the gravity-modified radiation acoustic mode exists with (#s 1)1 2 . For the particular parameters chosen, c;r c;diA . Right: Same as the left, but
for #T ¼ 3/2. Because #T is larger than unity, the gravitational instability is stabilized for all . Note the transition from adiabatic (radiation plus gas) to isothermal (gas only)
gravity-modified acoustic waves.
damped gravity-modified isothermal gas acoustic mode that
=
exists for k (#s #T )/2(#T 1)1 2 (dashed line).
Although I do not plot it here, equations (28)Y(32) and the
general solution to equation (27), show that in the limit #T ! 0
and #s ! 0 the medium is dynamically unstable for any .
decoupling. Because the total matter density (which sets the gravitational driving term, the numerator of eq. [33]) is roughly 10 times
the baryon density (which sets the scattering timescale, the denominator of eq. [33]), one expects a more careful derivation to
yield a timescale shorter by this ratio.
2.7. The Connection to Treatments of Silk Damping
2.8. Extensions
The calculation of the dispersion relation and the slow diffusion limit of x 2.3 is most closely related to the nonrelativistic
calculation of Silk (1967) in the context of the early universe. His
expression for the damping rate of adiabatic radiation-dominated
acoustic modes is essentially equivalent to the damping term in
equation (31), without the ‘‘1’’ in the denominator and with
#T ! 0. The correct relativistic expression for the damping rate
was subsequently obtained by Weinberg (1971). The qualitative
difference here with respect to Silk (1967) and other versions of
the derivation of the damping of acoustic modes in the early
universe by radiative diffusion (see, e.g., Hu & Sugiyama 1996)
is equation (32), which shows that the medium is unstable to a
slow diffusive mode precisely in the regime (#s > 1) where the
medium is stable to the adiabatic Jeans criterion—that is, on scales
below the adiabatic Jeans length (eq. [25]). A calculation of the
dispersion relation in the cosmological context (in an expanding
background) and an evaluation of the importance of this mode is
in preparation (T. A. Thompson 2008, in preparation). If it grows,
it should do so very slowly on a timescale many times the dynamical timescale, and it should be purely nonadiabatic. Although
2
¼ c 2 /3 in
I have not done a relativistic calculation here, taking cs;r
equation (33), I find that ! i4G/(c). This is the inverse of the
Kelvin-Helmholtz timescale for a relativistic radiation pressure
supported self-gravitating object.3 The characteristic timescale
for growth is of order tKH 1016 ð/0:4 cm2 g1 Þ s, which is of
order thousands of times longer than the age of the universe at
Motivated by Chandrasekhar (1961), Appendix A contains an
analysis analogous to x 2, but including uniform rotation in the
background medium. As in the case with only gas pressure, modes
propagating at right angles to the angular momentum vector are
stabilized by rotation if the angular velocity (
) is large enough
that 2 > G (see also Goldreich & Lynden-Bell 1965).
Appendix B accounts for the time dependence of the radiation
field and the possibility of imperfect energetic coupling between
the radiation field and the gas. For the parameters appropriate to
the astrophysical applications discussed in x 3 these factors are
largely unimportant for the qualitative stability properties of the
medium. This follows from the fact that the characteristic frequency for energetic coupling between the radiation and the gas,
!th c(ur /ug ) (see eq. [B8] and surrounding discussion, as
well as Bogdan et al. 1996; Blaes & Socrates 2003), is likely to be
large in the contexts considered. As in the work of, e.g., Dzhalilov
et al. (1992), Zhugzhda et al. (1993), and Bogdan et al. (1996), yet
more precise descriptions of the transport should be explored,
as should the dependence of the stability properties on the temperature and density dependence of the opacity (e.g., BisnovatyiKogan & Blinnikov 1979; Zhugzhda et al. 1993; Blaes & Socrates
2003) and the explicit dependence on the scattering albedo (e.g.,
Kaneko & Morita 2006).
Gradients in the background state—and, particularly, a background flux—have been neglected in the analysis of x 2. Ledoux
(1951) considered a consistent background state without invoking
the Jeans swindle in calculating the Jeans instability and found
only a small quantitative change to the stability properties of the
3
Note the correspondence with the Salpeter timescale for black hole growth.
No. 1, 2008
217
GRAVITATIONAL INSTABILITY
Fig. 2.—Left: Same as Fig. 1, but for #s ¼ 20 and #T ¼ 1/10, for 1 102 at fixed k 1 . Again, because #T < 1 on the scale chosen, the medium is unstable for
any 6¼ 0. For > c;diA #s (dashed line) the instability is dynamical, whereas for small < c;diA the growth rate for instability is (1 #T )/(#s 1) (eq. [32]).
The dotted line denotes c;r (eq. [38]). Right: Same as the left, but #T ¼ 3/2. Because #T is larger than unity, the medium is stable for all . The dashed line denotes the
=
critical above which the isothermal gas acoustic wave exists with (#T 1)1 2 .
pffiffiffi
medium: the Jeans length was increased by a factor of 2. A
detailed assessment of such gradients in the context of radiation
pressureYdominated self-gravitating media, as well as an exploration of magnetic fields and the physics of the photon bubble
instability (Arons 1992; Gammie 1998; Blaes & Socrates 2001,
2003) are saved for a future effort.
3. DISCUSSION
The analysis of x 2 indicates that under the assumption of
optically thick equilibrium radiative diffusion, an isotropic selfgravitating medium is unstable if the classical gas-only isothermal
Jeans criterion is not satisfied. If diffusion is rapid, the instability
is dynamical—the classical Jeans instability. If diffusion is slow,
then on scales larger than the adiabatic Jeans length the medium
is dynamically unstable—again, the Jeans instability. However,
on scales smaller than the adiabatic Jeans length the medium is
unstable to a diffusive mode that acts on a timescale of order the
Kelvin-Helmholz timescale (eq. [33]), longer than the dynamical
timescale of the medium.
Depending on whether or not the medium is rapidly or slowly
diffusing (cf. eqs. [39] and [40]), the growth time for gravitational instability may be significantly decreased with respect to
the dynamical timescale corresponding to the average density of
the medium. Here I discuss several astrophysical environments
where this analysis is applicable and where it provides some insight into the stability properties of the medium.
3.1. Radiation in Starbursts, ULIRGs, Parsec-Scale AGN
Disks, and Extreme Massive Star-forming Regions
Starburst galaxies are marked by high radiation energy density,
high gas density, and optical depths to their own dust-reprocessed
infrared radiation that exceed unity. Scoville et al. (2001), Scoville
(2003), and TQM05 have argued that radiation pressure may
dominate the dynamics in these systems. Typical temperatures
are in the range T 50Y200 K and densities range from n 500Y103 cm3 (e.g., the central regions of M82 and NGC 253)
to n k 2 ; 104 cm3 (e.g., the nuclei of the ULIRG Arp 220;
Downes & Solomon 1998) on 100 pc scales. In extreme massive star-forming regions temperatures are similar to those in
starburst galaxies and ULIRGs, but the typical density of clumps
and cores responsible for star formation can be higher. An example is the core of NGC 5253 with n 107 cm3 and a physical
scale of order 1 pc (e.g., Turner et al. 2000).
Yet more extreme physical conditions are expected to obtain
in the self-gravitating parsec-scale disks or obscuring ‘‘tori’’
thought to attend the process of fueling active galactic nuclei (e.g.,
Pier & Krolik 1994; Goodman 2003; Sirko & Goodman 2003;
TQM05). There, one expects central disk temperatures approaching the sublimation temperature of dust grains, Tsub 103 K.
Although the gas density is uncertain in these environments, we
can make an estimate by assuming that the disk is marginally
3
Toomre-stable such that n 2 /(2Gmp ) 6 ; 108 M8 R3
1pc cm ,
8
where M8 ¼ M /10 M and R1pc ¼ R/1 pc, within the sphere of
influence of the central supermassive black hole.4
Simple estimates indicate that the dust and gas in these systems
are collisionally and energetically coupled and that the IR optical
depth is larger than unity. Ignoring the enhancement of dust-gas
coupling due to magnetic fields and grain charging, the mean free
path for momentum coupling is
7 1
kdg 103 n1
4 pc 10 n8 pc;
ð41Þ
where nx ¼ n/10x cm3.5 The medium is optically thick to the
dust-reprocessed IR radiation field on scales larger than
1
k ¼1 ¼ ()1 81
2:5 n4 pc
1
8 ; 104 1
2:5 n8 pc;
4
ð42Þ
1=2 2
The sublimation radius for dust is Rsub 1L46
T3 pc, where T3 ¼ Tsub /103 K
1
46
and L46 ¼ LBH /10 ergs s is the Eddington luminosity for a 108 M black hole.
5
Here, I have assumed a dust grain density of 3 g cm3 and an average dust
grain radius of ad 0:1 m. Because grain charging and magnetic fields are
likely to be important the dust and gas may be regarded as a single, coupled fluid
on scales larger than kdg .
218
THOMPSON
where 2:5 ¼ /2:5 cm2 g1 is a representative Rosseland-mean
dust opacity for T 100 K, assuming solar metallicity and a
Galactic dust-to-gas ratio (e.g., Fig. 1 from Semenov et al. 2003).
For temperatures near the dust sublimation temperature, 2:5 1
is also a fair order-of-magnitude approximation (but see Chang
et al. 2007). Because k¼1 /kdg 8 ; 103 /2:5 3 1, the dust and
gas are always highly collisionally coupled if the average medium
is optically thick. A rough estimate of the optical depth in the
nuclei of the ULIRG Arp 220, where the scale of the system is
R 100 pc, is IR 102:5 n4 R100 pc . In systems like M82 and
NGC 253, n is lower and the medium is only marginally optically
thick on 100 pc scales. Although observations indicate that the obscuring material surrounding AGN may occupy a large fraction of
4, theoretical arguments suggest that most of the gas may be
confined to a thin disk with vertical scale height hTR (e.g.,
Thompson et al. 2005; Chang et al. 2007; Krolik 2007). The vertical
optical depth in such a disk is then IR 102 2:5 n8 (h/0:1 pc).
Although the dust-gas fluid is highly collisionally coupled on
the scales of interest, the energetic coupling may not be perfect.
However, in regions for which the cooling line radiation is optically thick, we do not expect large temperature differences between the gas and dust, similar to the case in dense molecular
clouds where the gas temperature is maintained by a combination
of heating by dust-gas collisions and cosmic rays, and cooling in
molecular lines (see, e.g., Gorti & Hollenbach 2004). At high
density, inelastic dust-gas collisions likely dominate gas heating.
Assuming order-unity differences between the gas and dust temperatures, the gas heating timescale is roughly
1=2
theat =tdyn 0:03n4
1=2
T2
1=2
104 n8
1=2
T3
;
ð43Þ
where here tdyn ¼ (G)1 2 . Assuming tight dust-gas coupling
and IR k 1, the medium is highly radiation pressure dominated:
=
2 3 1
pr =pg 103 T23 n1
4 10 T3 n8 :
ð44Þ
Although these estimates imply that starbursts and AGN disks
are optically thick and potentially modestly to very strongly radiation pressure dominated, the ratio6
(4G)1=2
1=2 4
14001
2:5 n4 T2
#s
(4pr =c)
1=2
4
141
2:5 n8 T3
ð45Þ
shows explicitly that diffusion of radiation in starbursts is very
‘‘rapid.’’ The stability properties of these media on the scale of
the system are thus best represented by the far right-hand portion
of the left panels of Figures 1 and 2: gas pressure is negligible
and radiation pressure is important as measured by the adiabatic
Jeans number #s, but 3 #s and the medium is dynamically
(Jeans) unstable.
For fiducial parameters, the scaling for AGN disks also indicates
that they are rapidly diffusing. However, the disk parameters in
this regime are quite uncertain. For example, Chang et al. (2007)
advocate 2:5 20 or larger for a gas density n 108 cm3 and
solar metallicity, implying /#s 0:7T34 . The linear dependence of on metallicity and the strong temperature dependence
of /#s implies that if grains persist for T k 103 K and/or the
composition of the disk is supersolar, the medium may transition
to slowly diffusing, and the timescale for gravitational instability
will be increased to tKH (eqs. [33] and [37]). Also, an estimate of
#T shows that gas pressure becomes important on small scales in
AGN disks and may stabilize the medium in the linear regime
(TQM05).
3.1.1. The Nonlinear Outcome
Consider an initial hypothetical equilibrium configuration for a
self-gravitating disk with starburst /ULIRG-like characteristics
such that #T T1, #s 1 on 100 pc scales and with cs2 2
pr / (h
)2 , where h is the disk scale height (as in the
cs;r
models of TQM05). Because diffusion is ‘‘rapid,’’ the configuration is dynamically unstable on all scales larger than the classical
gas Jeans length (Th) and vertical hydrostatic equilibrium cannot
be maintained in the linear regime.7
The nonlinear outcome of the Jeans instability in such a system
is highly uncertain in part because it is tied to star formation,
which, in turn, determines the character of the radiation field.
The question of whether or not hydrostatic equilibrium can be
maintained depends crucially on the nonlinear coupling of the
radiation and the gas. One possibility is that the large-scale radiation field produced by star formation is coupled to the generation
of turbulence, which regulates the structure of the galaxy and its
stability properties. Supersonic turbulence on large scales has
been shown to inhibit the Jeans instability and gravitational collapse (e.g., Klessen et al. 2000; Mac Low & Klessen 2004). It has
also recently been invoked as a basis for understanding the origin
of the Schmidt/Kennicutt laws (Kennicutt 1998; Krumholz &
McKee 2005). Indeed, the turbulent velocities inferred in local
starbursts and ULIRGs are large enough that a ‘‘turbulent Jeans
number,’’ #turb v2 k 2 /(4G), analogous to #s and #T , may
indicate marginal stability: #turb 1. Thus, if radiation pressure
forces can generate turbulence, perhaps a statistical hydrostatic
equilibrium can be maintained.
There are at least two reasons why—in the absence of energetic
input to the ISM from stars (e.g., supernovae)—star formation
may be coupled to the generation of turbulence. First, because
in the initial fictitious equilibrium state envisioned radiation pressure is large enough that pr / (h
)2 , the radiation field is capable of driving mass motions with velocities of order v h
if order-unity spatial variations in the radiation field are present.
Second, although I have not shown it in this paper, one expects
the astrophysical environments described here to be subject to
the self-gravitating analog of the photon bubble instability, which
in its nonlinear state will drive turbulence (Turner et al. 2005). The
latter is particularly interesting because it motivates a dynamical
coupling between the turbulent energy density (uturb ), the photon
energy density (uph ), and the magnetic energy density (uB ).
The second of these connections, between uph and uB in galaxies, can be motivated phenomenologically. Recently Thompson
et al. (2006) have shown that the magnetic field strengths in starbursts significantly exceed estimates derived from the ‘‘minimum
energy argument.’’ In addition, they show that uB must be a
constant, order unity, multiple of uph in these systems (see Condon
et al. 1991). This conclusion follows from the linearity of the
FIR-radio correlation, the radio spectral indices of star-forming
galaxies at GHz frequencies, and the fact that the ratio uB /uph
measures the importance of synchrotron versus inverse Compton
cooling of the cosmic-ray electrons and positrons (e.g., Condon
1992). The fact the galaxies that comprise the FIR-radio correlation have uph -values that span five to six dex, and that in the
Galaxy uph uB, implies that uB must increase with uph , from
normal Milky WayYlike galaxies to ULIRGs. The necessity of
7
6
Here, (A/CV ) in is taken as 1 (see eqs. [20] and [26]).
Vol. 684
The equilibrium imagined is likely also unstable to convective, magnetorotational, and photon-bubble instabilities (see Blaes & Socrates 2001, 2003).
No. 1, 2008
GRAVITATIONAL INSTABILITY
this lock-step increase in both uph and uB may signal a dynamical
coupling between the radiation field and the magnetic field in
galaxies. Thus, the fact that uph ; uB , and uturb are of the same order
of magnitude may not be a coincidence, but instead a necessary
consequence of the dynamical coupling between the radiation
field and the self-gravitating magnetized ISM.
3.2. Cosmic Rays in Normal Star-forming
Galaxies and Clusters
Although no attempt is made here to model the diffusion of
cosmic rays, it is instructive to consider the various parameters
governing gravitational stability in the case of cosmic rays vis à
vis radiation.
The total pressure in cosmic rays in the Galaxy is pcr; MW 1012 ergs cm3 (e.g., Boulares & Cox 1990), comparable to the
energy density in starlight, magnetic fields, and turbulence. The
cosmic-ray lifetime is inferred to be tcr (2Y3) ; 107 yr (GarciaMunoz et al. 1977; Connell 1998). Interpreted as a diffusion
timescale on kpc scales, one infers a cosmic-ray scattering mean
free path of order lmfp 0:1Y1 pc. In addition, from the observed grammage traversed by cosmic rays in the Galaxy, one
infers an average gas density encountered by the cosmic rays of
n 0:2 cm3 ( Engelmann et al. 1990). Writing lmfp ¼ ()1
2
(cf. eq. [45]; see also Kuwabara & Ko 2004), #s 50p12 n2
0:2 kkpc
and8
clmfp
3=2
(4G)1=2 0:4l0:1pc p1
cr;MW n0:2 :
#s
4pcr
ð46Þ
Although the parameters are uncertain, equation (46) indicates
that cosmic rays are marginally slowly diffusing in normal starforming galaxies. Thus, as for photons in the dense parsec-scale
AGN disks discussed in x 3.1, perhaps on kpc scales /#s may
be somewhat less than unity so that the growth rate for the
gravitational instability is (cf. eq. [37])
1=2
1
n0:2 :
tKH =tdyn 3pcr;MW l0:1pc
Depending on the phase of the ISM considered, the isothermal
Jeans number #T may be very close to unity so that the critical below which the Jeans instability is suppressed can be made
significantly larger (eq. [32]). Indeed, cosmic rays have recently
been proposed as an important large-scale feedback mechanism
in star-forming galaxies (Socrates et al. 2008).
Similar estimates may be written down for the central regions
of galaxy clusters, but it is unclear if these regions may plausibly
be cosmic-ray pressure dominated. If they are at least modestly
so, scaling from equation (46) for higher pressures and lower
densities, we see then that for 0:1 P lmfp P 100 pc they are plausibly in the ‘‘slow’’ diffusion limit (/#s < 1); again, the dynamical Jeans instability is quelled by the nonthermal pressure
support. In this limit, the diffusive instability identified in
equation (32) still acts on tKH , but this timescale is likely many
times the age of the universe: for cs; r 1000 km s1 and
n 102 , tKH 1012 (lmfp /pc) yr.
3.3. Individual Massive Stars
Individual massive stars are radiation pressure dominated
and slowly diffusing and may in principle also be subject to the
8
For simplicity, here I take A/CV ¼ 1 in . This is a rough approximation in
the context of the Galaxy because, depending on which gas phase of the ISM is
being considered, #T may be the same order of magnitude as #s .
219
diffusive mode identified in x 2. This is simply a secular instability of the kind discussed in, e.g., Hansen (1978, and references therein). If present, the growth timescale is the Kelvin2
/(4Gc) 103 yr for typical
Helmholz time, of order tKH 3cs;r
parameters, where is the Thomson opacity and cs;r is the adiabatic radiation pressureYdominated sound speed of the fluid
=
(GM /R)1 2 . Although massive stars are known to be globally
secularly unstable on the Kelvin-Helmholtz timescale, it is possible that otherwise stably stratified (radiative) regions of their
interiors may be locally unstable to a variant of the diffusive
instability in equation (33).
4. SUMMARY AND CONCLUSION
I consider the physics of gravitational instabilities in the presence of dynamically important radiation pressure and radiative
diffusion. I find that the medium is always stable on scales smaller
than the gas-only isothermal Jeans length, kJ;T (eq. [23]). For
scales larger than kJ;T there are two possibilities, depending on
whether the medium is ‘‘slowly’’ or ‘‘rapidly’’ diffusing, as
defined in x 2.5. When diffusion is rapid, radiation leaks out of a
perturbation without providing a sufficient restoring pressure
force and the medium is dynamically unstable on all scales larger
than kJ;T , regardless of the dominance of radiation pressure.
The limit of slow diffusion is more interesting. Here, the
medium is unstable to a diffusive mode at an intermediate range
of scales between the gas-only isothermal Jeans length kJ;T and
the larger (gas + radiation) adiabatic Jeans length kJ;s (eq. [25]).
The characteristic growth timescale is longer than the dynamical
timescale. Neglecting gas pressure, it is given approximately by
equation (33) (see also eqs. [32] and [37]), which is simply the
Kelvin-Helmholz timescale for a radiation pressureYsupported
self-gravitating fluid parcel to radiate its total thermal energy at
the Eddington limit. Note that on small spatial scales, the characteristic timescale is independent of scale. For k > kJ;s the
medium is dynamically ‘‘Jeans’’ unstable, as expected. Thus,
even when radiation pressure is dynamically dominant, on precisely
the scales where the medium is dynamically stable by the usual
Jeans criterion (k < kJ;s ), it is unstable to a diffusive instability
that operates on the Kelvin-Helmholtz time. I conclude that
radiation cannot formally stabilize a self-gravitating medium on
scales larger than the gas-only isothermal Jeans length. See also
the discussion of Kaneko & Morita (2006).
In x 3.1, I consider the importance of the results derived in x 2
for extreme sites of massive star formation including starburst
galaxies and parsec-scale AGN disks. I argue that the average
medium in these systems is likely to be radiation pressure dominated and optically thick. Importantly, for fiducial parameters the
photons in these systems are in the rapidly diffusing limit (/#s >
1; eq. [45]). For fairly extreme choices for the uncertain physical
parameters in parsec-scale AGN disks (e.g., the opacity ) this
environment is marginally slowly diffusing and thus the stability
properties of the medium might be qualitatively different from
rapidly diffusing starbursts.
TQM05 developed a theory of marginally Toomre-stable
starburst and AGN disks supported by feedback from radiation
pressure. Because the IR photons produced by dust-reprocessed
starlight in these systems diffuse rapidly and because their characteristic sizes are much larger than the classical gas-only Jeans
length, the analysis presented here dictates that they cannot be
supported in the linear regime by radiation pressure alone. One
may wonder, then, why the entire mass of gas in starbursts does
not fragment into stars on a single dynamical time, in apparent
contradiction with observations (e.g., Kennicutt 1998). If radiation pressure is to be the dominant feedback mechanism, then the
220
THOMPSON
answer must be that these forces are coupled to the generation of
supersonic turbulence, which may mitigate against complete collapse and fragmentation on scales larger than the gas-only Jeans
length. In x 3.1.1 I argue that the generation of turbulence likely
proceeds from the nonlinear coupling of the Jeans instability with
the radiation field through star formation, and may be driven by
the self-gravitating analog of the photon bubble instability. This
may help explain the apparent order-of-magnitude equivalence
between the radiation, magnetic, and turbulent energy densities
in starburst systems. Thus, it is important to emphasize that the
conclusion that radiation pressure alone cannot stave off gravitational instability in the rapidly diffusing limit does not necessarily
imply that the disk cannot be maintained in global hydrostatic
equilibrium in an average sense by radiation pressure in the nonlinear, turbulent regime.
In x 3.2, I consider the case of cosmic rays diffusing in the
Galaxy and the cosmic-ray halo, and in galaxy clusters. Although
no attempt is made here to calculate the physics of cosmic-ray
diffusion and their thermal coupling to the gas, they provide a
useful point of contrast with radiation because of their very high
Vol. 684
scattering optical depths. Even so, this constituent of the ISM of
the Galaxy is at the border of ‘‘slow’’ and ‘‘rapid’’ diffusion
(/#s 1) outlined in x 2 (eq. [46]) for fiducial parameters. In
the cluster context diffusion is likely more fully in the ‘‘slow’’
limit, but it is unclear if cosmic rays dominate the total pressure
budget in the central regions (e.g., Guo & Oh 2008). Finally, I
also briefly mention the possibility that individual massive stars
may be locally unstable to this diffusive mode on the local KelvinHelmholz timescale in otherwise stably stratified radiative regions
of their interiors.
This paper was motivated in part by stimulating conversations
with Aristotle Socrates. I also thank Yoram Lithwick, Andrew
Youdin, Charles Gammie, Kristen Menou, and Jeremy Goodman
for several useful conversations and Eliot Quataert, Norm Murray,
Bruce Draine, and Julian Krolik for encouragement. Finally, I am
grateful to the Department of Astrophysical Sciences at Princeton
University, where much of this work was completed. This paper
is dedicated to Garnett A. B. Thompson.
APPENDIX A
UNIFORM ROTATION
Chandrasekhar (1954, 1961) explored the effect of uniform rotation on the Jeans instability and found that the dispersion relation is
modified by the Coriolis force in the rotating frame. In particular, he showed that for the special case of waves propagating at right
ˆ ( cos2 ¼ 0) that equation (1) becomes ! 2 ¼ 4
2 þ c 2 k 2 4G—that is, if 2 > G, then the Jeans
angles to the direction g
6 0—for waves whose wavevectors
instability is stabilized for any gas sound speed cg (pg ¼ cg2 assumed). For the general case cos2 ¼
have arbitrary angles with respect to the spin axis—if the classical gas Jeans criterion (eq. [2]) indicates instability (#T < 1), then the
medium is unstable for any j
j. In order to gain some intuition and to make contact with the work of Chandrasekhar (1954) it is useful
to consider the Jeans instability including uniform rotation, radiation pressure, and radiative diffusion. The equation expressing conservation of momentum in the rotating frame is
@v
1
þ v = :v ¼ :P : þ 2(v < 6):
@t
ðA1Þ
All other equations in the original analysis of x 2 are unchanged. I take k ¼ (0; 0; kz ) and 6 ¼ (0; y ; z ). The perturbation equations
in component form are
i! þ ikz vz ¼ 0;
i!vx 2vy z þ 2vz y ¼ 0;
i!vy þ 2vx z ¼ 0;
i!vz þ ikz P= þ ikz 2vx y ¼ 0;
i!U þ (U þ P)ikz vz þ ikz Fz ¼ 0;
k 2 4G ¼ 0;
ikz ur þ 3Fz =c ¼ 0:
The resulting dispersion relation is (cf. eq. [19])
˜ 4 ! 3 4
2 þ cs2 k 2 4G i!!
˜ 2 4
2 þ cT2 k 2 4G
! 5 þ i!!
2
˜
cos2 ) cT2 k 2 4G ¼ 0;
þ !(4
2 cos2 ) cs2 k 2 4G þ i!(4
ðA2Þ
where cos ¼ z /j
j and !˜ is the radiation diffusion rate given in equation (17). Defining
Q ¼ 2 =(G);
ðA3Þ
and using the definitions for #T , #s , , and , in equations (22)Y(26), equation (A2) can be rewritten as
5 þ i4 3 ðQ þ #s 1Þ i 2 ðQ þ #T 1Þ þ Q cos2 ð#s 1Þ þ iQ cos2 ð#T 1Þ ¼ 0:
ðA4Þ
No. 1, 2008
221
GRAVITATIONAL INSTABILITY
For Q ¼ 0, equation (A4) reduces to equation (27). In addition, for the special case cos2 ¼ 0 equation (A4) reduces to equation (27)
with the substitutions #T ! Q þ #T and #s ! Q þ #s . Therefore, for Q 1 and cos2 ¼ 0, the medium is stabilized for any , #T ,
and #s . As in Chandrasekhar (1961), I find that for all cos2 6¼ 0, if #T < 1, the medium is unstable.
APPENDIX B
MORE GENERAL TREATMENTS OF RADIATION TRANSPORT
The prescription for radiation transport in x 2 makes several approximations. In particular, it neglects the time dependence of the
radiation field and it assumes perfect radiative equilibrium so that the radiation and gas temperatures are identical. The latter assumption
is particularly suspect when diffusion is rapid on a scale k 1 , since radiative equilibrium may not be possible to maintain. In fact, contrary to
the results of x 2, when the radiation and gas temperatures are distinguished, the gas acoustic speed in the limit of rapid diffusion should be
the adiabatic gas sound speed and not the isothermal gas sound speed (e.g., Mihalas & Mihalas 1984). More detailed treatments of radiating
flows without self-gravity may be found in Dzhalilov et al. (1992), Zhugzhda et al. (1993), and Bogdan et al. (1996). Kaneko & Morita
(2006) provide a detailed treatment of the radiation that distinguishes between scattering and pure absorptive opacity.
For completeness, here I present an analysis similar to x 2, but including the dynamics of the radiation field and allowing for
energetic decoupling between the radiation and gas. The set of equations is (cf. eqs. [3]Y[7])
D
þ : = v ¼ 0;
Dt
Dv 1
þ :pg þ : F=c ¼ 0;
Dt Dug
þ ug : = v c ur aT 4 ¼ 0;
Dt
Dur 4
þ ur : = v þ : = F þ c ur aT 4 ¼ 0;
3
Dt
92 4G ¼ 0;
1 DF c
þ :ur þ F ¼ 0;
c Dt
3
ðB1Þ
where D/Dt ¼ @ /@t þ v = :v is the Lagrangian derivative, T is the gas temperature, a ¼ 4SB /c is the radiation energy density constant,
and is the adiabatic index of the gas. The time derivative of the flux in the Euler equation has been neglected. The perturbation equations are
i! þ ik = v ¼ 0;
i!v þ ik(pg =) þ ik F=c ¼ 0;
i!ug þ ug ik = v þ c(ur Ag T ) ¼ 0;
i!ur þ (4=3)ur ik = v þ ik = F c(ur Ag T ) ¼ 0;
k 2 4G ¼ 0;
i!F þ c 2 =3 ikur þ (c)F ¼ 0:
ðB2Þ
where I have taken (aT 4 ) ¼ 4aT 3 T ¼ Ag T and employed the Jeans swindle. The perturbations to the gas pressure and energy density
are written as pg ¼ cT2 þ (@pg /@T )j T and ug ¼ cT2 /( 1) þ (@ug /@T )j T . Solving equation (B2), the resulting dispersion
relation can be written in a number of ways. The form most conducive to comparison with equation (19) is perhaps
4 ur 2
k 4G
!5 þ i !4 (2 þ s) ! 3 2 (1 þ s) þ 2 þ cT2 k 2 þ
9 4s
4 ur 2
i ! 2 2 s þ cT2 k 2
þ2 þ
k ð3 2 þ sÞ (2 þ s)4G
3
9 2
4s
4 ur 2
2
þ 2 ! cT2 k 2
þ1þ 2 þ
k ð3 3 þ sÞ 4G 1 þ s þ 2
3
9 2 2 2
ðB3Þ
þ i s cT k 4G ¼ 0;
where
s¼
Ag
4aT 4
¼
;
@ug =@T j
ug
2 ¼
c 2k 2
;
3
¼ c;
2
2
¼
1
1
1 1
¼
:
3 (k 1 )2 3 k2
ðB4Þ
222
THOMPSON
Vol. 684
The latter is the inverse of the optical depth squared across a scale k 1 . With G ¼ 0, equation (B3) is identical to equation (101.62) of
Mihalas & Mihalas (1984).9 For large optical depth in a radiation pressureYdominated medium, 2 / 2 T1Ts in the third and fifth terms
of equation (B3). In addition, for a nonrelativistic medium 2 3 (4ur /9)k 2 , cT2 k 2 , and 4G in the fourth term in equation (B3). Using
this ordering, dividing through by the quantity 2 s 2, and noting that 2 / ¼ ck 2 /(3) (cf. eqs. [17] and [26]), equation (B3) becomes
4
!5
!
ck 2 2
4 2 2 4 ur 2
ck 2 2 2
3
c
!
k
cT k 4G 0:
þ
i
!
k
þ
4G
i
þ
!
T
2 2
3
9 i s
3
3
s
ðB5Þ
This expression should be compared with equation (19). Note that the last four terms in equation (B5) are qualitatively identical to the
terms in equation (19) in the radiation pressureYdominated limit (A/CV ) 1. The importance of the first and second terms in equation (B5)
are measured by the characteristic frequency s with respect to the wave frequencies the expression admits. Generically, for sufficiently
large s/!, these terms are subdominant. Thus, if these limits hold, the qualitative stability properties outlined in x 2 obtain.
It is simplest to understand the origin of the extra terms in equation (B3) with respect to equation (19) by taking a step back.
Neglecting the time dependence of the flux in the last expression in equation (B1), but leaving the rest of the above analysis unchanged, I
find that
2
4 ur 2
4
3
2
2
2 2
k 4G
! þ i ! 1 þ s þ 2 ! s þ cT k þ
9 2
4 ur 2
2
2 2 4s
i ! cT k
þ1þ 2 þ
k ð3 3 þ sÞ 4G 1 þ s þ 2 þ s 2 cT2 k 2 4G ¼ 0: ðB6Þ
3
9 The somewhat peculiar terms multiplying cT2 k 2 and (4/9)ur k 2 / in the fourth term of the dispersion relation (e.g., 4s/3) are made
clear by examining the explicit and full expression for the sound speed of the radiation and the gas, at constant total entropy (cf. eq. [18]).
When ur / 3 cT2 , one finds that cs2 (4/3)cT2 þ (4/9)ur /, whereas when ur /TcT2 , one finds that cs2 cT2 þ (4/3)( 1)ur /. This
shows that for large and small s, these terms reduce to the adiabatic sound speed for the radiation and the gas, respectively. As in deriving
equation (B5), if I take 2 / 2 T1Ts (second and fourth terms) and 2 3 (4ur /9)k 2 , cT2 k 2 , and 4G (third term), I find that equation
(B6) can be written simply as
!4
ck 2 2
4 2 2 4 ur 2
ck 2 2 2
3
cT k þ
! !
k 4G i
c k 4G ¼ 0:
þ! þi
3
9 i s
3
3 T
ðB7Þ
Compare with equation (B5). The importance of s is again evident. As emphasized by Bogdan et al. (1996) and Blaes & Socrates
(2003), the characteristic frequency
4aT 4
!th ¼ s ¼ c
ðB8Þ
ug
measures the rate at which energy is exchanged between the matter and the radiation field. When this frequency is large, the energetic
coupling is tight and the analysis presented in x 2 is recovered. Thus, for very large s and vanishingly small diffusion rate across a scale
k 1 — 2 / ¼ ck 2 /(3) ! 0—only the second and third terms in equation (B7) survive: ! 3 !(4cT2 k 2 /3 þ (4/9)ur k 2 / 4G) 0.
That is, in the optically thick limit with slow diffusion the radiation pressure contributes to the stability of the system against gravitational
collapse. As in x 2, the fact that it appears that the system is stabilized if (4/9)ur k 2 / > 4G, even when (4/3)cT2 k 2 < 4G is an artifact
of taking the limit of zero diffusion rate. Taking just the last two terms in equation (B6) (the small ! limit) and then taking 2 / 2 ! 0
(high optical depth on a scale k 1 ) and then s ! 1 (for tight energetic coupling between the radiation and the matter, radiation pressure
dominated), I find that
! i
2
cT2 k 2 4G
ck 2
1 #T
¼
i
;
3 (4=3)#T þ #r 1
(4=3)cT2 k 2 þ (4=9)(ur =)k 2 4G
ðB9Þ
where
#r ¼
4 ur k 2
1
9 (4G)
ðB10Þ
is defined in analogy with #T and #s (eqs. [22] and [24]). This expression should be compared with equation (32); in the radiation pressureY
dominated limit they are identical. Thus, as in x 2, I find that even for highly radiation pressureYdominated media, with very large optical
depth and strong energetic coupling between matter and radiation, the medium is unstable if cT2 k 2 < 4G (the isothermal Jeans number
#T < 1). As before, at high-k, the characteristic timescale for instability is independent of spatial scale and is simply the Kelvin-Helmholz
timescale (cf. eqs. [33] and [37]).
9
Correcting for a sign error in the first term of their eq. (101.58).
No. 1, 2008
Alternatively, taking the limit
223
GRAVITATIONAL INSTABILITY
2
! 0 in equation (B6), so that 2 /
3 s, 1, and 2 s equation (B6) becomes
! 3 !(cT2 k 2 4G) 0:
ðB11Þ
Contrary to the discussion of x 2, which showed that in the limit of rapid diffusion the gas acoustic mode speed is the isothermal sound
speed cT , here I find that when the energetic coupling between the radiation and the gas is weak, acoustic modes propagate at the
adiabatic sound speed 1=2 cT , as expected (e.g., Mihalas & Mihalas 1984). This effect was not accounted for in the analysis of x 2
because perfect energetic coupling was assumed. Equation (B11) shows that in the limit ! 0, the classical gas Jeans criterion is
obtained and that on scales larger than the Jeans length, the medium is unstable, even if (4/9)ur k 2 / 3 4G. However, equation
(B11) is somewhat deceiving as it may imply to the reader that the medium can be stabilized in the special case cT2 k 2 < 4G, but
cT2 k 2 > 4G in the ! 0 limit. This is false, and an artifact of having taken ¼ 0 in obtaining equation (B11). Expanding instead
to first order in , I find the unstable mode is
cT2 k 2 4G
1 #T
! i s
¼i s
;
#T 1
cT2 k 2 4G
ðB12Þ
which shows that in the special case cT2 k 2 < 4G, but cT2 k 2 > 4G, the medium is unstable.
Expanding equation (B6) in the high-k limit I find that
! 1=2 cT k i
3
2
4 ur
s 1
;
þ
9 c 2
3
ðB13Þ
in agreement with Blaes & Socrates (2003, their eq. [57]). In a nonrelativistic medium, the second term in square brackets dominates
so that ! 1=2 cT k i s( 1)/2. Thus, in the high-k limit gas acoustic waves are damped by emission and absorption, again
with characteristic damping rate s. Although in equation (B13) I obtain a wave speed equal to the adiabatic gas sound speed, as in
equation (B11), the high-k limit is not identical to the limit ! 0 because the gravitational term, which dictates stability/instability,
disappears at high k. To see this, I write ! ¼ (cT2 k 2 4G)1=2 þ iq in equation (B6), take only linear terms in q, and then expand
to first order as ! 0. I find that
!
(cT2 k 2
4G)
1=2
3
i
2
4 ur
s ( 1)cT2 k 2
þ
:
9 c 2
3 cT2 k 2 4G
ðB14Þ
For large k, equation (B14) reduces to equation (B13). However, on scales where gravity is important, equation (B14) shows that if
cT2 k 2 k 4G then the damping rate of gravity-modified adiabatic gas acoustic waves is altered from the prediction of equation
(B13). More importantly, we see explicitly that if cT2 k 2 < 4G the acoustic mode is unstable.
Taken together, equations (B12) and (B14) show that in the limit of small s, (1) if both #T < 1 and #T < 1, then the medium is
unstable, and (2) that if #T < 1 and #T > 1, then the medium is also unstable. In addition, equation (B9) shows that when s is very
large and the energetic coupling between matter and radiation is tight, the medium is also unstable for #T < 1, even if #r 31, as in x 2.
Thus, one expects that the medium is only globally stable on a scale k 1 if the classical Jeans criterion is satisfied and the isothermal
Jeans number is larger than unity, #T > 1.
B1. FULL SOLUTION TO THE DISPERSION RELATION
In analogy with #T , #s , , and #r (eqs. [22]Y[26] and [B10]), I define the quantities
¼
c 2k 2
1
2
¼
;
3 (4G) (4G)
¼
s
(4G)1=2
;
ðB15Þ
where measures the rate of thermal coupling between radiation and gas in units of the dynamical timescale. With these definitions,
the approximate solutions and limits of the previous subsection are illustrated in Figure 3, which presents the full solution to equation (B6)
over a broad range of / / k1 for the parameters ¼ 107 , ¼ 5/3, #r ¼ 3, #T ¼ 1/5 (top left), #T ¼ 1/2 (top right), #T ¼ 4/5
(bottom left), and #T ¼ 3/2 (bottom right) at fixed scale k 1 . In each panel, I take s ¼ 9( 1)#r /#T so that s ¼ 180, 36, 22.5, and 12
from left to right, top to bottom, respectively. As in Figures 1 and 2, open and filled circles show the real and imaginary components of
=
¼ !/(4G)1 2 . Positive complex components indicate unstable modes. Note that in each panel the damping rates i and
i are off-scale at intermediate values of / .
In each panel, because #r —and, by extension, #s —is larger than unity, in the limit of large k (large , small / ) solutions
qualitatively identical to those obtained in Figures 1 and 2 are recovered. Thus, the left-hand portions of all panels are similar to Figure 1.
The qualitatively new feature of this figure is the small-k regions in each panel. In the top two panels both cT2 k 2 and cT2 k 2 are less than
4G (#T < 1 and #T < 1) so that the medium is unstable for any / . For sufficiently small / , the growth rate is subdynamical and
224
THOMPSON
Vol. 684
Fig. 3.—Solution to eq. (B6) for ¼ 107 , ¼ 5/3, #r ¼ 3, and #T ¼ 1/10 (top left), and for #T ¼ 1/2 (top right), #T ¼ 4/5 (bottom left), #T ¼ 3/2 (bottom right), for
a very wide range of / ¼ k1 /31=2 at fixed scale k 1 . Open and filled circles show the real and imaginary parts of the roots , respectively. Positive complex roots
indicate instability. Components of the dispersion relation are labeled for clarity. The left portion of each panel (large k , strong thermal coupling) is qualitatively similar
to Figs. 1 and 2. In each panel #r > 1 so that #s > 1. The bottom left panel shows the special case #T < 1 and #T > 1. At very large / , the gravity- and radiationmodified adiabatic gas only acoustic wave exists and in this region the growth rate for the Jeans instability is suppressed.
equal to the inverse of the Kelvin-Helmholz timescale (cf. eqs. [33] and [37]). Here, the diffusive instability operates. However, for
=
intermediate values of the / , the growth rate is dynamical [ i(1 #T )1=2 ], whereas for very large / ; i(1 #T )1 2 . In
these cases, the medium is classically Jeans unstable.
The bottom left panel is different. Here, #T < 1, but #T > 1 (see eq. [B12]). In this special case, two sets of gravity- and diffusionmodified acoustic waves exist: (1) at small / , the adiabatic radiation pressure dominated acoustic waves are evident (as in the top two
panels) and (2) at large / , adiabatic gas (only) acoustic waves are also present. Note that in this regime (small thermal coupling, small k )
the medium is still unstable, but the growth rate for instability is (1 #T )(#T 1)1 (eq. [B12]) in the special case where both the
numerator and the denominator are positive.
The bottom right panel shows a case analogous to the right panels of Figures 1 and 2 with #T > 1. Here, the Jeans instability is stabilized
=
=
at all k . At intermediate / ; (#T 1)1 2 , whereas for large values of / , (#T 1)1 2 . As in the other panels, in the limit
1=2
of strong thermal coupling between the radiation and the gas (large k , small / ), (#s 1) .
No. 1, 2008
GRAVITATIONAL INSTABILITY
225
B2. SCALINGS AND APPLICATIONS
The qualitative differences at small optical depth or poor thermal coupling between the radiation and the matter in the right-hand
portion of each of the panels in Figure 3 with respect to Figures 1 and 2 do not change the conclusions about most of the astrophysical
applications discussed in x 3 because fiducial estimates for and are very large:
s
c
4aT 4
1=2
1=2
¼
108 2:5 n4 T23 109 2:5 n8 T33
¼
ug
(4G)1=2
(4G)1=2
c 12( 1) pcr
8 1
1=2 pcr
¼
l
n
7
;
10
;
ðB16Þ
0:1pc
lmfp (4G)1=2 pg
pg
¼
2
c 2 k 2 =3
c 2
1
8 2 1
¼
7 ; 106 k2
2 n4 7 ; 10 k1 n8
4G
4G
Gk2
1
7 ; 108 k2
kpc n :
ðB17Þ
In addition, the ratio / is
¼
1
31=2
k
1
2
1
1 1
¼ 1=2
0:32:5 n1
4 k2 0:032:5 n8 k1
k
3
2lmfp
1
¼ 1=2
4 ; 104 l0:1pc k1
kpc :
k
3
ðB18Þ
The first line of each equation shows the scalings for starbursts and AGN disks, while the second line shows the scaling appropriate for
cosmic rays (cf. eq. [46]). For all cases considered, is very large and / is less than unity. The stability properties of these media are
thus best represented by the left-hand portion of each of the panels in Figure 3; the radiation and the matter are tightly energetically coupled.
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